| L(s) = 1 | + 2.12·2-s − 2.07·3-s + 2.52·4-s + 5-s − 4.41·6-s − 2.06·7-s + 1.11·8-s + 1.31·9-s + 2.12·10-s − 3.02·11-s − 5.24·12-s − 3.05·13-s − 4.39·14-s − 2.07·15-s − 2.67·16-s + 4.38·17-s + 2.79·18-s − 6.63·19-s + 2.52·20-s + 4.29·21-s − 6.43·22-s − 3.12·23-s − 2.31·24-s + 25-s − 6.49·26-s + 3.50·27-s − 5.21·28-s + ⋯ |
| L(s) = 1 | + 1.50·2-s − 1.19·3-s + 1.26·4-s + 0.447·5-s − 1.80·6-s − 0.780·7-s + 0.393·8-s + 0.438·9-s + 0.672·10-s − 0.912·11-s − 1.51·12-s − 0.846·13-s − 1.17·14-s − 0.536·15-s − 0.669·16-s + 1.06·17-s + 0.659·18-s − 1.52·19-s + 0.564·20-s + 0.936·21-s − 1.37·22-s − 0.651·23-s − 0.472·24-s + 0.200·25-s − 1.27·26-s + 0.673·27-s − 0.985·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 985 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 2 | \( 1 - 2.12T + 2T^{2} \) |
| 3 | \( 1 + 2.07T + 3T^{2} \) |
| 7 | \( 1 + 2.06T + 7T^{2} \) |
| 11 | \( 1 + 3.02T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 4.38T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 + 3.12T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 + 2.54T + 37T^{2} \) |
| 41 | \( 1 - 0.608T + 41T^{2} \) |
| 43 | \( 1 - 0.159T + 43T^{2} \) |
| 47 | \( 1 + 2.10T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 12.0T + 59T^{2} \) |
| 61 | \( 1 + 12.7T + 61T^{2} \) |
| 67 | \( 1 + 2.00T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 + 7.46T + 79T^{2} \) |
| 83 | \( 1 - 3.76T + 83T^{2} \) |
| 89 | \( 1 - 3.88T + 89T^{2} \) |
| 97 | \( 1 - 15.4T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07979031004884323758084217941, −8.794270691560939456401477312153, −7.44594973974257426650751326391, −6.46121268395021709076258787750, −5.96198087329977503405027498211, −5.24219743289421419719728567675, −4.55835807077951667784965791626, −3.32368269521355796336341618976, −2.30933043483793532323442798682, 0,
2.30933043483793532323442798682, 3.32368269521355796336341618976, 4.55835807077951667784965791626, 5.24219743289421419719728567675, 5.96198087329977503405027498211, 6.46121268395021709076258787750, 7.44594973974257426650751326391, 8.794270691560939456401477312153, 10.07979031004884323758084217941
